Homogenization Expansion for Resonances of Microstructured Photonic Waveguides

We develop a homogenization expansion approach to photonic waveguides, whose transverse structures are N-fold rotationally symmetric. Examples include microstructured, or "holey," optical fibers with air holes arranged in one or more concentric rings. Our methods apply to structures, where the microfeatures have arbitrary geometry. We carry out a homogenization/perturbation expansion for large N about the N = infinity limit. We are interested in structures possessing leaky modes, and in computing the properties of these modes. The leading order (N = infinity) equations describe the modes of an averaged structure. We derive an expansion in powers of 1/N of corrections to the leading order behavior, and show that the lowest nontrivial contribution arises at order 1/N sup 2. We numerically calculate this leading order correction to the complex effective indices (scattering resonances) for the leaky modes of various nmicrostructured photonic waveguides, whose imaginary parts give the leakage rates.